Overview:  In this Launch, teacher performs a review of similar triangles before students begin their work on the problem.  The teacher reminds students that they have probably studied similar figures in an earlier grade, and she reminds students that similar figures are the same shape, but not necessarily the same size.  The teacher draws two triangles on the board, next to each other, and asks students to decide whether they are similar.  Students cannot determine whether the triangles are similar, because there is no way to relate them.  Next, the teacher draws a triangle inside one of the original triangles, sharing an angle.  The teacher also tells students that the sides opposite the shared angle are congruent.  Then students decide the two triangles are similar.  The teacher tells students that they do not have to use similar figures to solve the shadow puppet problem, but when they see triangles in their diagrams they may want to look for relationships between them.

Prior knowledge:  The teacher elicits students’ prior knowledge of the school mathematics concept of similarity.  Assuming that students have studied similar figures in an earlier grade, the teacher uses examples to review similar versus non-similar figures.  The teacher does not give a formal definition of similarity during this launch.  Instead she gives a vague definition—“figures that are the same in some ways, but not necessarily exactly the same”—and the teacher uses one non-example and one example to illustrate the mathematical concept.  The students in the Launch seem to have some prior knowledge of “similar,” suggesting that it could refer to things that are “the same” or “the same but not exactly the same.”  From students’ comments throughout the Launch, it is unclear whether students are thinking about similarity as a formal mathematical term or whether students are drawing upon their informal prior experiences with things that are similar (Cox, Lo, & Mingus, 2007; Lehrer, Strom, & Confrey, 2002).

Other points of interest:  The teacher seems to intend to review the definition of similarity through the use of examples and non-examples.  However, the teacher’s choice of diagrams for the review do not necessarily provide a clear definition of mathematical similarity.  There are no angle or side markings on either of the first two triangles the teacher presents.  Without any markings, one could not assume anything about the relationships between the two triangles.  Because the triangles are drawn with angles that are close to equal, and with almost the same orientation, the triangles look similar, although they are not.  The “almost similar” triangles may cause even more confusion given the informal definition of similarity that the class is using.  When the teacher adds a third triangle, nested inside the first triangle, she seems to intend to illustrate a pair of similar triangles, taking for granted that the sides are parallel.  What the teacher refers to ask a “hint”—marking the sides of the two triangles parallel—is actually a necessary condition for the triangles to be similar.  The Launch raises a question about the best choice of diagrams and what norms the teacher establishes about marking or making assumptions about diagrams.